Homotopy type of manifolds with partially horoconvex boundary

Abstract

Let $M$ be an $n$-dimensional compact connected manifold with boundary, $\kappa >0$ a constant and $1\le q\le n-1$ an integer. We prove that $M$ supports a Riemannian metric with the interior $q$-curvature $K_q\ge -q{\kappa }^2$ and the boundary $q$-curvature ${\Lambda }_q\ge q\kappa$, if and only if $M$ has the homotopy type of a CW complex with a finite number of cells with dimension $\le (q-1)$. Moreover, any Riemannian manifold $M$ with sectional curvature $K\ge -{\kappa }^2$ and boundary principal curvature $\Lambda \ge \kappa$ is diffeomorphic to the standard closed $n$-ball.

11 pages; accepted by Internat. J. Math